| L(s) = 1 | − 2·2-s + 4-s − 4·7-s + 2·8-s − 2·11-s + 8·14-s − 4·16-s + 4·17-s − 12·19-s + 4·22-s + 4·23-s − 4·28-s − 12·29-s − 8·31-s + 2·32-s − 8·34-s + 32·37-s + 24·38-s − 2·41-s − 10·43-s − 2·44-s − 8·46-s − 4·47-s + 12·49-s − 24·53-s − 8·56-s + 24·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.51·7-s + 0.707·8-s − 0.603·11-s + 2.13·14-s − 16-s + 0.970·17-s − 2.75·19-s + 0.852·22-s + 0.834·23-s − 0.755·28-s − 2.22·29-s − 1.43·31-s + 0.353·32-s − 1.37·34-s + 5.26·37-s + 3.89·38-s − 0.312·41-s − 1.52·43-s − 0.301·44-s − 1.17·46-s − 0.583·47-s + 12/7·49-s − 3.29·53-s − 1.06·56-s + 3.15·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05660265936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05660265936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 10 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 - 20 T^{2} + 231 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T - 10 T^{2} + 80 T^{3} - 221 T^{4} + 80 p T^{5} - 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 80 p T^{5} - 8 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T - 5 T^{2} + 190 T^{3} + 4876 T^{4} + 190 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 76 T^{2} - 8 T^{3} + 5503 T^{4} - 8 p T^{5} - 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 2 T + 35 T^{2} + 298 T^{3} - 2756 T^{4} + 298 p T^{5} + 35 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 8 p T^{5} - 104 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T + 19 T^{2} + 602 T^{3} + 13708 T^{4} + 602 p T^{5} + 19 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 136 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 104 T^{2} + 4575 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.85934787301664589727153291607, −6.74042445853328884785243528904, −6.52014629898770229784946468461, −6.27508099964263242612481768377, −6.01857063879957646581834825497, −5.85713378690841019442814341864, −5.66368460758130718116018695604, −5.34662542325445546372725553863, −5.05946821199069051270039225008, −5.04212092178310158361949310413, −4.45064048363137946074938506038, −4.25877510313444359586355744192, −4.03436958633878684338116184850, −3.97720622262018784983392353855, −3.84117635950114504206095144400, −3.02165608148900063760570956212, −3.00547304902650612275748777160, −2.87188775840963021039364430964, −2.70152647730484150154313557091, −2.04290339753211947038474179905, −1.89435082893973505463307238468, −1.44811142962811307830475593414, −1.28895602039534303826382008075, −0.51270757216369847913234352106, −0.10709028405462893851947625571,
0.10709028405462893851947625571, 0.51270757216369847913234352106, 1.28895602039534303826382008075, 1.44811142962811307830475593414, 1.89435082893973505463307238468, 2.04290339753211947038474179905, 2.70152647730484150154313557091, 2.87188775840963021039364430964, 3.00547304902650612275748777160, 3.02165608148900063760570956212, 3.84117635950114504206095144400, 3.97720622262018784983392353855, 4.03436958633878684338116184850, 4.25877510313444359586355744192, 4.45064048363137946074938506038, 5.04212092178310158361949310413, 5.05946821199069051270039225008, 5.34662542325445546372725553863, 5.66368460758130718116018695604, 5.85713378690841019442814341864, 6.01857063879957646581834825497, 6.27508099964263242612481768377, 6.52014629898770229784946468461, 6.74042445853328884785243528904, 6.85934787301664589727153291607
